The Solution of Embedding Problems in the Framework of GAPs with Applications on Nonlinear PDEs

Table 1

Symmetry	Definition	Corresponding𝐶(∗)-symmetry
(anti)commutativity	𝑥∗𝑦=(−)𝑦∗𝑥	𝐶^𝑘(∗)𝑖𝑗=(−)𝐶^𝑘(∗)𝑗𝑖
associativity	(𝑥∗𝑦)∗𝑧=𝑥∗(𝑦∗𝑧)	𝐶^𝑙(∗)𝑖𝑗𝐶^𝑚(∗)𝑙𝑘=𝐶^𝑙(∗)𝑗𝑘𝐶^𝑚(∗)𝑖𝑙
Jacobi symmetry	(𝑥∗𝑦)∗𝑧+(𝑦∗𝑧)∗𝑥	𝐶^𝑙(∗)𝑖𝑗𝐶^𝑚(∗)𝑙𝑘+𝐶^𝑙(∗)𝑗𝑘𝐶^𝑚(∗)𝑙𝑖
	+(𝑧∗𝑥)∗𝑦=0	+𝐶^𝑙(∗)𝑘𝑖𝐶^𝑚(∗)𝑙𝑗=0
𝑒(∗,𝐿)-existence	𝑒(∗,𝐿)∗𝑥=𝑥	𝑒^𝑖(∗,𝐿)𝐶^𝑘(∗)𝑖𝑗=𝛿^𝑘𝑗
𝑒(∗,𝑅)-existence	𝑥∗𝑒(∗,𝑅)=𝑥	𝐶^𝑘(∗)𝑖𝑗𝑒^𝑗(∗,𝑅)=𝛿^𝑘𝑗